Proof that class of languages accepted by DPDA by empty stack is not closed under union

Question

My first intuition was to take two languages $L_1$ and $L_2$ (symbol $d$ at the end is to fulfill prefix property):

$$L_1 = \{ a^i b^i c^j d : i,j \ge 0 \} \mathrm{\ \ and\ \ } L_2 = \{ a^i b^j c^j d : i,j \ge 0 \}$$

So their union would be: $ L = L_1 \cup L_2 = \{ a^i b^i c^k d : i=j \lor j=k \} $

Then to prove $L$ not even being DCFL (e.g. in similar way as Example 10.1 from "Introduction to Automata Theory, Languages, and Computation" by Hopcroft & Ullman).

Big inconvenience would be the need to include in the proof also the proofs of all other properties (assuming this class is - just as "regular" DCFL - closed under complement etc.)

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But I was told that precisely because the question is about DCFL being accepted by empty stack, the proof is actually trivial and could be expressed in just few sentences.

Answer 1

Indeed, the requirement that the DPDA accepts by empty stack has some "severe" consequences.

Under the standard definitions, an automaton with empty stack is blocked from further computations, as normally one would pop a stack single symbol at each computational step. As a consequence DPDA languages by empty stack are prefix-free: if the accepted language contains a string $x$ then it will not contain any string $xy$ with $y\neq \varepsilon$.

Now the language $\{a,aa\}$ is an easy counterexample.